Algebraic Division

During world examination flavour I'm afraid I can't resist looking at The Student Room - non to read all the infuriating educatee posts where they obsess virtually lost marks in addition to aeroplane boundaries, but merely to run into each examination newspaper every bit presently every bit possible. I receive got no patience. Arsey, presumably an anonymous teacher, posts model answers real presently later the exams receive got taken house (in previous years non until the side past times side day, but at i time in that place are unlike papers for unlike fourth dimension zones the model answers are posted direct later the Great Britain sitting). When I was reading Arsey's recent C2 solutions, I noticed his method for polynomial division:

I've seen him do this before. It ever occurs to me that this is far to a greater extent than straightforward than the long partitioning method I learn my students. Why do I learn long division?  I suppose it's because it features inward the C2 textbook thus I've ever assumed it's the 'best' method. Perhaps side past times side yr I'll endeavor an alternative. In this post service I await at 4 methods for polynomial division. 

1. Long Division

This method often features inward H5N1 aeroplane textbooks. It exactly involves next a serial of steps (divide, multiply, subtract, select down, repeat) - an algorithm learnt past times drill rather than through understanding. The steps are familiar to those who learnt long partitioning at primary school. Those who were taught choice partitioning methods (eg chunking) are at a slight disadvantage but do grab upward quickly. Practice makes perfect. This isn't an elegant method - it's totally procedural in addition to isn't specially prissy to teach, in addition to students receive got to know special rules for special cases (eg including a 0x term) - but it does the labor exactly fine. It industrial plant good when there's a remainder.

2. Grid/Box Method
I've exactly tried this method for the kickoff fourth dimension in addition to I can't believe how slow it is - in addition to thus much quicker than long division! All y'all receive got to do is ready a multiplication grid - start past times filling inward the bits y'all know in addition to and then the relaxation follows past times logic. This video from Bon Crowder explains the method real clearly. James Tanton calls this the Galley Method - his Curriculum Essay about how it industrial plant includes exercises in addition to interesting questions. The film below from Esther (@MrsMathematica) shows a sensible layout which makes dealing amongst remainders real easy.

3. Inspection
This is like the grid method but laid out differently. All y'all receive got to do is write your polynomial every bit the production of a linear purpose in addition to an unknown quadratic (or cubic, quartic etc, depending on the question) in addition to then occupation logic in addition to algebra to operate out the numbers past times equating the coefficients. It's quick in addition to fairly straightforward. It's also easier to follow what's going on than inward the confusing algorithm of long division.

In a 1940s textbook I spotted an alternative layout for inspection that I love. In the instance below nosotros desire to split upward past times x - 1 thus nosotros write (x - 1) 3 times in addition to and then exactly fill upward inward the rest. Again, it's quick in addition to logical, in addition to slow to cash inward one's chips on rail of each term.

4. Synthetic Division

I don't similar this method so I don't genuinely desire to get upward it here, but for completeness I suppose I should. I've constitute lots of (often negative) reference to it on American websites but I've never seen it used inward England. I'm told that it's usually used in Scotland (thanks to @mrallanmaths, @kenniejp23 in addition to Paul Smith for commenting). The argue I dislike it is it appears to hold out i of those 'remember the steps but receive got no clue what's genuinely happening' methods.

Source: acedemic.utep.edu

I followed the rules higher upward in addition to it did work, addition it was pretty quick, but it was a fighting similar doing a magic trick.

The animation below shows the equivalence of long partitioning in addition to synthetic division. It looks to me that synthetic partitioning is exactly a confusing method made fifty-fifty to a greater extent than abstract. There are many defenders of synthetic partitioning though - they tell that it's an acceptable method providing students are taught the underlying concept earlier they start applying the super-efficient algorithm.

Source: purplemath.com

So that's it - 4 methods for dividing polynomials. This PowerPoint from the Further Mathematics Support Programme summarises the kickoff 3 methods.

Which do y'all prefer?